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  <div class="section" id="numpy-random-randomstate-pareto">
<h1>numpy.random.RandomState.pareto<a class="headerlink" href="#numpy-random-randomstate-pareto" title="Permalink to this headline">¶</a></h1>
<p>method</p>
<dl class="method">
<dt id="numpy.random.RandomState.pareto">
<code class="sig-prename descclassname">RandomState.</code><code class="sig-name descname">pareto</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">size=None</em><span class="sig-paren">)</span><a class="headerlink" href="#numpy.random.RandomState.pareto" title="Permalink to this definition">¶</a></dt>
<dd><p>Draw samples from a Pareto II or Lomax distribution with
specified shape.</p>
<p>The Lomax or Pareto II distribution is a shifted Pareto
distribution. The classical Pareto distribution can be
obtained from the Lomax distribution by adding 1 and
multiplying by the scale parameter <code class="docutils literal notranslate"><span class="pre">m</span></code> (see Notes).  The
smallest value of the Lomax distribution is zero while for the
classical Pareto distribution it is <code class="docutils literal notranslate"><span class="pre">mu</span></code>, where the standard
Pareto distribution has location <code class="docutils literal notranslate"><span class="pre">mu</span> <span class="pre">=</span> <span class="pre">1</span></code>.  Lomax can also
be considered as a simplified version of the Generalized
Pareto distribution (available in SciPy), with the scale set
to one and the location set to zero.</p>
<p>The Pareto distribution must be greater than zero, and is
unbounded above.  It is also known as the “80-20 rule”.  In
this distribution, 80 percent of the weights are in the lowest
20 percent of the range, while the other 20 percent fill the
remaining 80 percent of the range.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>New code should use the <code class="docutils literal notranslate"><span class="pre">pareto</span></code> method of a <code class="docutils literal notranslate"><span class="pre">default_rng()</span></code>
instance instead; see <em class="xref py py-obj">random-quick-start</em>.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>a</strong><span class="classifier">float or array_like of floats</span></dt><dd><p>Shape of the distribution. Must be positive.</p>
</dd>
<dt><strong>size</strong><span class="classifier">int or tuple of ints, optional</span></dt><dd><p>Output shape.  If the given shape is, e.g., <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">n,</span> <span class="pre">k)</span></code>, then
<code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span> <span class="pre">*</span> <span class="pre">k</span></code> samples are drawn.  If size is <code class="docutils literal notranslate"><span class="pre">None</span></code> (default),
a single value is returned if <code class="docutils literal notranslate"><span class="pre">a</span></code> is a scalar.  Otherwise,
<code class="docutils literal notranslate"><span class="pre">np.array(a).size</span></code> samples are drawn.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">ndarray or scalar</span></dt><dd><p>Drawn samples from the parameterized Pareto distribution.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lomax.html#scipy.stats.lomax" title="(in SciPy v1.4.1)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.stats.lomax</span></code></a></dt><dd><p>probability density function, distribution or cumulative density function, etc.</p>
</dd>
<dt><a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.genpareto.html#scipy.stats.genpareto" title="(in SciPy v1.4.1)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.stats.genpareto</span></code></a></dt><dd><p>probability density function, distribution or cumulative density function, etc.</p>
</dd>
<dt><a class="reference internal" href="numpy.random.Generator.pareto.html#numpy.random.Generator.pareto" title="numpy.random.Generator.pareto"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Generator.pareto</span></code></a></dt><dd><p>which should be used for new code.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>The probability density for the Pareto distribution is</p>
<div class="math">
<p><img src="../../../_images/math/9676b3d3f7cb08b4e71d965108d0809dfc4d5c15.svg" alt="p(x) = \frac{am^a}{x^{a+1}}"/></p>
</div><p>where <img class="math" src="../../../_images/math/b3e65e3b6408fcfa00452530b73f55d1755f9965.svg" alt="a"/> is the shape and <img class="math" src="../../../_images/math/e9bc7da808d33a16a8347f27a519bd067186aa66.svg" alt="m"/> the scale.</p>
<p>The Pareto distribution, named after the Italian economist
Vilfredo Pareto, is a power law probability distribution
useful in many real world problems.  Outside the field of
economics it is generally referred to as the Bradford
distribution. Pareto developed the distribution to describe
the distribution of wealth in an economy.  It has also found
use in insurance, web page access statistics, oil field sizes,
and many other problems, including the download frequency for
projects in Sourceforge <a class="reference internal" href="#r4338a4b3d731-1" id="id1">[1]</a>.  It is one of the so-called
“fat-tailed” distributions.</p>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r4338a4b3d731-1"><span class="brackets"><a class="fn-backref" href="#id1">1</a></span></dt>
<dd><p>Francis Hunt and Paul Johnson, On the Pareto Distribution of
Sourceforge projects.</p>
</dd>
<dt class="label" id="r4338a4b3d731-2"><span class="brackets">2</span></dt>
<dd><p>Pareto, V. (1896). Course of Political Economy. Lausanne.</p>
</dd>
<dt class="label" id="r4338a4b3d731-3"><span class="brackets">3</span></dt>
<dd><p>Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme
Values, Birkhauser Verlag, Basel, pp 23-30.</p>
</dd>
<dt class="label" id="r4338a4b3d731-4"><span class="brackets">4</span></dt>
<dd><p>Wikipedia, “Pareto distribution”,
<a class="reference external" href="https://en.wikipedia.org/wiki/Pareto_distribution">https://en.wikipedia.org/wiki/Pareto_distribution</a></p>
</dd>
</dl>
<p class="rubric">Examples</p>
<p>Draw samples from the distribution:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="mf">3.</span><span class="p">,</span> <span class="mf">2.</span>  <span class="c1"># shape and mode</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">pareto</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">m</span>
</pre></div>
</div>
<p>Display the histogram of the samples, along with the probability
density function:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">count</span><span class="p">,</span> <span class="n">bins</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">fit</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">m</span><span class="o">**</span><span class="n">a</span> <span class="o">/</span> <span class="n">bins</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">bins</span><span class="p">,</span> <span class="nb">max</span><span class="p">(</span><span class="n">count</span><span class="p">)</span><span class="o">*</span><span class="n">fit</span><span class="o">/</span><span class="nb">max</span><span class="p">(</span><span class="n">fit</span><span class="p">),</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<div class="figure align-default">
<img alt="../../../_images/numpy-random-RandomState-pareto-1.png" src="../../../_images/numpy-random-RandomState-pareto-1.png" />
</div>
</dd></dl>

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